* http://www.danielsoper.com/statcalc/calc23.aspx
*
* It calculates upper incomplete gamma function, which equals
- * kf_gammap(s,z)*tgamma(s).
+ * kf_gammaq(s,z)*tgamma(s).
*/
#define KF_GAMMA_EPS 1e-14
double C, D, f;
f = 1. + z - s; C = f; D = 0.;
// Modified Lentz's algorithm for computing continued fraction
- // See Numerical Recipes in C, 2nd edition, page 172
+ // See Numerical Recipes in C, 2nd edition, section 5.2
for (j = 1; j < 100; ++j) {
double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
D = b + a * D;
return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
}
+/* Regularized incomplete beta function. The method is taken from
+ * Numerical Recipe in C, 2nd edition, section 6.4. The following web
+ * page calculates the incomplete beta function, which equals
+ * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
+ *
+ * http://www.danielsoper.com/statcalc/calc36.aspx
+ */
+static double kf_betai_aux(double a, double b, double x)
+{
+ double C, D, f;
+ int j;
+ if (x == 0.) return 0.;
+ if (x == 1.) return 1.;
+ f = 1.; C = f; D = 0.;
+ // Modified Lentz's algorithm for computing continued fraction
+ for (j = 1; j < 200; ++j) {
+ double aa, d;
+ int m = j>>1;
+ aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
+ : m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
+ D = 1. + aa * D;
+ if (D < KF_TINY) D = KF_TINY;
+ C = 1. + aa / C;
+ if (C < KF_TINY) C = KF_TINY;
+ D = 1. / D;
+ d = C * D;
+ f *= d;
+ if (fabs(d - 1.) < KF_GAMMA_EPS) break;
+ }
+ return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
+}
+double kf_betai(double a, double b, double x)
+{
+ return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
+}
+
#ifdef KF_MAIN
#include <stdio.h>
int main(int argc, char *argv[])
{
double x = 5.5, y = 3;
+ double a, b;
printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
- printf("lower-gamma(%lg,%lg): %lg\n", x, y, (1.0-kf_gammap(y, x))*tgamma(y));
+ printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
+ a = 2; b = 2; x = 0.5;
+ printf("incomplete-beta(%lg,%lg,%lg): %lg\n", a, b, x, kf_betai(a, b, x) / exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b)));
return 0;
}
#endif